Can the Lanczos algorithm converge very fast by taking a good initial guess?

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Suppose I have the two lowest eigenvectors $v_1$, $v_2$ of a matrix $M$.

If slightly change $M$ to $M'$. Can I use $v_1$ or $v_2$ as an initial guess for $M'$?

If so, which one should be used, $v_1$ or $v_2$? (suppose I want the two lowest eigenvectors of the new matrix $M'$)